Heat and Chemical Transport in Aquifers at Different Geological Setting

Heat and chemical transport are two fundamental processes that are widely existed in the subsurface environments and come with natural phenomena (e.g., volcanic eruption, diurnal or seasonal temperature variation) or anthropogenic activities (e.g., well injection). A common characteristic of these two transport processes is they are both governed by the advection-dispersion equation (ADE) and both advective movements are impacted by the local heterogeneity of porous media. However, they still exhibit many different behaviors. For example, heat diffusivity in the solid matrix is usually two order of magnitude higher than the matrix diffusivity of chemicals. In addition, chemical transport usually comes with reaction and sorption. To investigate the characteristic behaviors of heat and chemical transport at different geological settings, this study focuses on three geological environments ranging from kilometer-scale to meter-scale, including volcanic hydrothermal system, shallow riverbeds where surface-water and groundwater exchange occurs and local fractured aquifer for well testing. For the volcanic hydrothermal system, a novel model that connects the heat and chemical transport is proposed to explain the over 10-year temperature and chemical data collected at the thermal springs near volcanic summit. For the shallow riverbeds, an ensemble data assimilation approach is proposed to estimate the hydraulic exchange flux between surface water and groundwater based on the heat transport observed in the riverbeds. For the local fractured aquifer, a novel fractional model for single-well push-pull test is proposed to explain the observed long tailing behavior of conservative tracer during pumping. This study demonstrates the capability of using heat and chemical as tracers to quantitatively or qualitatively estimate the flow and transport behaviors at different geological environments. Further work is needed to explore the capability of model and methods to accommodate more geological conditions

Elimination of the reaction rate 'scale effect': application of the Lagrangian reactive particle-tracking method to simulate mixing-limited, field-scale biodegradation at the Schoolcraft (MI, USA) site

This is the peer reviewed version of the following article: [Ding, D., Benson, D. A., Fernàndez‐Garcia, D., Henri, C. V., Hyndman, D. W., Phanikumar, M. S., & Bolster, D. (2017). Elimination of the reaction rate “scale effect”: Application of the Lagrangian reactive particle‐tracking method to simulate mixing‐limited, field‐scale biodegradation at the Schoolcraft (MI, USA) site. Water Resources Research, 53, 10,411–10,432. https://doi.org/10.1002/2017WR021103], which has been published in final form at https://doi.org/10.1002/2017WR021103. This article may be used for non-commercial purposes in accordance with Wiley Terms and Conditions for Self-Archiving.Measured (or empirically fitted) reaction rates at groundwater remediation sites are typically much lower than those found in the same material at the batch or laboratory scale. The reduced rates are commonly attributed to poorer mixing at the larger scales. A variety of methods have been proposed to account for this scaling effect in reactive transport. In this study, we use the Lagrangian particle-tracking and reaction (PTR) method to simulate a field bioremediation experiment at the Schoolcraft, MI site. A denitrifying bacterium, Pseudomonas Stutzeri strain KC (KC), was injected to the aquifer, along with sufficient substrate, to degrade the contaminant, carbon tetrachloride (CT), under anaerobic conditions. The PTR method simulates chemical reactions through probabilistic rules of particle collisions, interactions, and transformations to address the scale effect (lower apparent reaction rates for each level of upscaling, from batch to column to field scale). In contrast to a prior Eulerian reaction model, the PTR method is able to match the field-scale experiment using the rate coefficients obtained from batch experiments.Peer ReviewedPostprint (author's final draft

Boundary Conditions for Fractional Diffusion

This paper derives physically meaningful boundary conditions for fractional diffusion equations, using a mass balance approach. Numerical solutions are presented, and theoretical properties are reviewed, including well-posedness and steady state solutions. Absorbing and reflecting boundary conditions are considered, and illustrated through several examples. Reflecting boundary conditions involve fractional derivatives. The Caputo fractional derivative is shown to be unsuitable for modeling fractional diffusion, since the resulting boundary value problem is not positivity preserving

Numerical solution of variable-order time fractional weakly singular partial integro-differential equations with error estimation

In this paper, we apply Legendre-Laguerre functions (LLFs) and collocation method to obtain the approximate solution of variable-order time-fractional partial integro-differential equations (VO-TF-PIDEs) with the weakly singular kernel. For this purpose, we derive the pseudo-operational matrices with the use of the transformation matrix. The collocation method and pseudo-operational matrices transfer the problem to a system of algebraic equations. Also, the error analysis of the proposed method is given. We consider several examples to illustrate the proposed method is accurate

Application of fractional sub-equation method to nonlinear evolution equations

In this paper, we constructed a traveling wave solutions expressed by three types of functions, which are hyperbolic, trigonometric, and rational functions. By using a fractional sub-equation method for some space-time fractional nonlinear partial differential equations (FNPDE), which are considered models for different phenomena in natural and social sciences fields like engineering, physics, geology, etc. This method is a very effective and easy to investigate exact traveling wave solutions to FNPDE with the aid of the modified Riemann–Liouville derivative

Numerical solution of fractional partial differential equations by spectral methods

Fractional partial differential equations (FPDEs) have become essential tool for the modeling of physical models by using spectral methods. In the last few decades, spectral methods have been developed for the solution of time and space dimensional FPDEs. There are different types of spectral methods such as collocation methods, Tau methods and Galerkin methods. This research work focuses on the collocation and Tau methods to propose an efficient operational matrix methods via Genocchi polynomials and Legendre polynomials for the solution of two and three dimensional FPDEs. Moreover, in this study, Genocchi wavelet-like basis method and Genocchi polynomials based Ritz- Galerkin method have been derived to deal with FPDEs and variable- order FPDEs. The reason behind using the Genocchi polynomials is that, it helps to generate functional expansions with less degree and small coefficients values to derive the operational matrix of derivative with less computational complexity as compared to Chebyshev and Legendre Polynomials. The results have been compared with the existing methods such as Chebyshev wavelets method, Legendre wavelets method, Adomian decomposition method, Variational iteration method, Finite difference method and Finite element method. The numerical results have revealed that the proposed methods have provided the better results as compared to existing methods due to minimum computational complexity of derived operational matrices via Genocchi polynomials. Additionally, the significance of the proposed methods has been verified by finding the error bound, which shows that the proposed methods have provided better approximation values for under consideration FPDEs